Problem 11 - Precalculus Student Solutions Manual 5th

Question

Explain how to determine whether to use a parenthesis or a square bracket when graphing the solution set of a linear inequality.

Step-by-Step Solution

Answer

Use a parenthesis for '<' and '>', and a square bracket for '≤' and '≥'.

Step 1 - Understanding the Inequality Symbols

Identify the inequality symbol used in the given linear inequality. The symbols '<' and '>' indicate strict inequalities, meaning the boundary point is not included in the solution set. The symbols '≤' and '≥' indicate inclusive inequalities, meaning the boundary point is included.

Step 2 - Deciding on Parenthesis or Bracket

Use a parenthesis '()' if the boundary point is not included in the solution set. This corresponds to the strict inequalities '<' and '>'. Use a square bracket '[]' if the boundary point is included in the solution set. This corresponds to the inclusive inequalities '≤' and '≥'.

Step 3 - Plotting on a Number Line

Graph the solution set on a number line. If you use a parenthesis, the point is open, indicating that it is not part of the solution set. If you use a square bracket, the point is closed, indicating that it is part of the solution set.

Key Concepts

Inequality Symbols

Inequality symbols are crucial in determining how we represent the boundaries of our solutions. There are two main types:

• Strict inequalities: These are represented by the symbols '<' (less than) and '>' (greater than). They mean the value at the boundary is not included in the solution set.
• Inclusive inequalities: These are shown with the symbols '≤' (less than or equal to) and '≥' (greater than or equal to). Here, the boundary value is included in the solution set.

Understanding these differences is key as they dictate how we visualize the solutions on a graph.
The strict inequalities '<' and '>' require that the endpoint is not included, while the inclusive inequalities '≤' and '≥' ensure the endpoint is included.

Parenthesis vs. Square Bracket

When graphing linear inequalities, choosing between a parenthesis or a square bracket is important. Here’s how to decide:

• Parenthesis '()': This is used when the boundary point is not part of the solution set. It corresponds to the symbols '<' and '>'. If the inequality is strict, you will use a parenthesis.
• Square Bracket '[]': This is applied when the boundary point is included in the solution set. It relates to the symbols '≤' and '≥'. For inclusive inequalities, a square bracket indicates the endpoint is part of the solution.

To summarize, parenthesis means the boundary is excluded, and square brackets mean the boundary is included.

Number Line Plotting

Plotting the solution set on a number line helps visual learners understand the boundaries and range of solutions. Here’s a step-by-step guide:

• Identify the key boundary point from your inequality.
• Decide if this point should be included, based on your symbol (use a square bracket if it's included or a parenthesis if it's not).
• Mark this point on your number line. If it’s included, make a solid dot. If it’s excluded, make an open circle.
• Shade the portion of the number line that represents the solution set.

For example, if your inequality is 'x > 3', you would place an open circle at 3 and shade to the right. If it’s 'x ≤ -2', you would place a solid dot at -2 and shade to the left. This visual representation makes understanding solutions more intuitive.